He was the first-born child and only son of the Upper Silesian railway entrepreneur and coal mine owner Rudolf Pringsheim (1821–1901) and his wife Paula, née Deutschmann (1827–1909).

Pringsheim considered himself to be a German citizen who no longer followed the "Mosaic belief" (meaning conservative or orthodox Judaism).

His first-born son, Erik, was exiled to Argentina because of his dissolute life and gambling debts and died there at an early age.

His sons Peter and Klaus followed him in pursuing academic careers, obtaining professorships in physics and musical composition.

One musician in the family was enough, so his third son, Heinz, became an archaeologist with a doctorate in that field, but soon changed course, becoming a successful conductor and critic in Berlin and Munich.

In 1889 Pringsheim and his family moved into a Neo-Renaissance villa at Arcisstrasse 12 designed by the Berlin architects Kayser & von Großheim with interior furnishings provided by Joh.

Besides mathematics, ever since his youth Pringsheim was also intensively occupied with music, and adapted various compositions of Richard Wagner for the piano.

[4] Pringsheim published numerous works on the subject of complex analysis, with a focus on the summability theory of infinite series and the boundary behavior of analytic functions.

One of Pringsheim's theorems, according to Hadamard[5] earlier proved by E. Borel, states[6] that a power series with positive coefficients and radius of convergence equal to 1 has necessarily a singularity at the point 1.

[10][11] Pringsheim and Ivan Śleszyński, working separately, proved what is now called the Śleszyński–Pringsheim theorem on convergence of certain continued fractions.

Besides his research in analysis, Pringsheim also wrote articles for the Enzyklopädie der mathematischen Wissenschaften on the fundamentals of arithmetic and on number theory.

He also had to sell his marvellous mathematics library which contained many precious books dating back to the sixteenth century.

One of many antisemitic pieces of Nazi legislation, the Namensänderungsverordnung [de] which came into effect 1 January 1938, forced him to legally change his name into Alfred Israel Pringsheim at age 87.

Through the intervention of the then-rector of Munich University (LMU), his former neighbor Karl Haushofer, who was a friend of Rudolf Hess, and the professor of mathematics Oskar Perron, one of Alfred Pringsheim's former students, as well as through the initiative of a courageous member of the SS who arranged for passports at the last minute, he and his wife were able to leave for Zürich, Switzerland on 31 October 1939 after suffering further grave humiliations.

[18] His world famous collection of majolica was sold in a forced sale by the Nazis at Sotheby's in London in 1939 in exchange for permission to emigrate.

His wife then apparently burned all of the personal effects which had been brought to Switzerland, including the letters from Richard Wagner.

Pringheim's heirs have requested that artworks looted by the Nazis and sold in forced sales be returned to the family.